## thomas5267 one year ago Prove that the polynomials $$P_n$$ has n distinct root for all n. $$P_n$$ are characteristic polynomials of a particular type of matrix. $P_n=-xA_{n-1}-\frac{1}{2}A_{n-2}\\ A_n = \left( \dfrac{-x+\sqrt{x^2-1}}{2}\right)^n + \left(\dfrac{-x-\sqrt{x^2-1}}{2}\right)^n$ $$A_n$$ satisfies the following recurrence relation: $A_n=-x A_{n-1}-\frac{1}{4}A_{n-2}\\ A_1=-x\\ A_2=x^2-\frac{1}{2}$

1. anonymous

im not sure if we need to find th characteristic coefficient and show they are distinct

2. anonymous

@zzr0ck3r

3. zzr0ck3r

no idea

4. thomas5267

I want to prove these matrices are diagonalisable. $M_5=\begin{pmatrix} 0&\frac{1}{2}&0&0&0\\ 1&0&\frac{1}{2}&0&0\\ 0&\frac{1}{2}&0&\frac{1}{2}&0\\ 0&0&\frac{1}{2}&0&1\\ 0&0&0&\frac{1}{2}&0 \end{pmatrix}\\ M_7=\begin{pmatrix} 0&\frac{1}{2}&0&0&0&0&0\\ 1&0&\frac{1}{2}&0&0&0&0\\ 0&\frac{1}{2}&0&\frac{1}{2}&0&0&0\\ 0&0&\frac{1}{2}&0&\frac{1}{2}&0&0\\ 0&0&0&\frac{1}{2}&0&\frac{1}{2}&0\\ 0&0&0&0&\frac{1}{2}&0&1\\ 0&0&0&0&0&\frac{1}{2}&0 \end{pmatrix}\\$ In order words, $$(M_n)_{i,i+1}=(M_n)_{i+1,i}=\frac{1}{2}$$ except for $$(M_n)_{2,1}=(M_n)_{n-1,n}=1$$

5. thomas5267

The old question has gotten so convoluted I figured that I will post a new one.

6. zzr0ck3r

This is just an algorithm...

7. anonymous

seems diagonalizable , but look at the diagonal zero which might have zero and its determinant might be zero in both and not being singular not sure though i can't help. maybe @Empty

8. zzr0ck3r

find the eigen values and the eigen vectors

9. thomas5267

As far as I can check, from $$M_3$$ to $$M_{15}$$ the matrices all have distinct eigenvalues.

10. zzr0ck3r

ahh you are trying to show for the whole family?

11. thomas5267

Yes. Favard's theorem seems to apply in here. Those polynomials should be orthogonal.

12. anonymous

well, there is a reason why i hate orthogonality. i'll watch and learn!

13. anonymous

@dan815 @Kainui @Empty when ever your free just lets try on this!

14. thomas5267

I know nothing about Favard's theorem but I found this seemingly useful theorem on google.

15. thomas5267

I have no idea whether $$A_n$$ is orthogonal or not. It does not seem like the case.

16. Empty

Can you use the Cayley Hamilton theorem? It says that every matrix satisfies its characteristic equation.

17. thomas5267

What I want to prove is that the matrices $$M_n$$ is always diagonalisable. A sufficient but not necessary condition is that the characteristic polynomials of $$M_n$$ have n distinct roots. Not sure how Cayley Hamilton Theorem will help though.